Xem mẫu
RF and Microwave Wireless Systems. Kai Chang Copyright # 2000 John Wiley & Sons, Inc.
ISBNs: 0-471-35199-7 (Hardback); 0-471-22432-4 (Electronic)
CHAPTER TWO
Review of Waves and Transmission Lines
2.1 INTRODUCTION
At low RF, a wire or a line on a printed circuit board can be used to connect two electronic components. At higher frequencies, the current tends to concentrate on the surface of the wire due to the skin effect. The skin depth is a function of frequency and conductivity given by
1=2
ds oms 2:1
where o 2pf is the angular frequency, f is the frequency, m is the permeability, and s is the conductivity. For copper at a frequency of 10GHz, s 5:8107 S=m and d 6:6 105 cm, which is a very small distance. The ®eld amplitude decays exponentially from its surface value according to ez=ds, as shown in Fig. 2.1. The ®eld decays by an amount of e1 in a distance of skin depth ds. When a wire is operating at low RF, the current is distributed uniformly inside the wire, as shown in Fig. 2.2. As the frequency is increased, the current will move to the surface of the wire. This will cause higher conductor losses and ®eld radiation. To overcome this problem, shielded wires or ®eld-con®ned lines are used at higher frequencies.
Many transmission lines and waveguides have been proposed and used in RFand microwave frequencies. Figure 2.3 shows the cross-sectional views of some of these structures. They can be classi®ed into two categories: conventional and integrated circuits. A qualitative comparison of some of these structures is given in Table 2.1. Transmission lines and=or waveguides are extensively used in any system. They are used for interconnecting different components. They form the building blocks of
10
2.1 INTRODUCTION 11
FIGURE 2.1 Fields inside the conductor.
FIGURE 2.2 The currrent distribution within a wire operating at different frequencies.
many components and circuits. Examples are the matching networks for an ampli®er and sections for a ®lter. They can be used for wired communications to connect a transmitter to a receiver (Cable TV is an example).
The choice of a suitable transmission medium for constructing microwave circuits, components, and subsystems is dictated by electrical and mechanical trade-offs. Electrical trade-offs involve such parameters as transmission line loss, dispersion, higher order modes, range of impedance levels, bandwidth, maximum operating frequency, and suitability for component and device implementation. Mechanical trade-offs include ease of fabrication, tolerance, reliability, ¯exibility, weight, and size. In many applications, cost is an important consideration.
This chapter will discuss the transmission line theory, re¯ection and transmission, S-parameters, and impedance matching techniques. The most commonly used transmission lines and waveguides such as coaxial cables, microstrip lines, and rectangular waveguides will be described.
12 REVIEW OF WAVES AND TRANSMISSION LINES
FIGURE 2.3 Transmission line and waveguide structures.
2.2 WAVE PROPAGATION
Waves can propagate in free space or in a transmission line or waveguide. Wave propagation in free space forms the basis for wireless applications. Maxwell predicted wave propagation in 1864 by the derivation of the wave equations. Hertz validated Maxwell`s theory and demonstrated radio wave propagation in the
13
14 REVIEW OF WAVES AND TRANSMISSION LINES
laboratory in 1886. This opened up an era of radio wave applications. For his work, Hertz is known as the father of radio, and his name is used as the frequency unit.
Let us consider the following four Maxwell equations:
H E e Gauss` law 2:2a
H E @t Faraday`s law 2:2b
H H @t J Ampere`s law 2:2c H B 0flux law 2:2d
where E and B are electric and magnetic ®elds, D is the electric displacement, H is the magnetic intensity, J is the conduction current density, e is the permittivity, and r is the charge density. The term @D=@t is displacement current density, which was ®rst added by Maxwell. This term is important in leading to the possibility of wave propagation. The last equation is for the continuity of ¯ux.
We also have two constitutive relations:
D e0E P eE 2:3a
B m0H M mH 2:3b
where P and M are the electric and magnetic dipole moments, respectively, m is the permeability, and e is the permittivity. The relative dielectric constant of the medium and the relative permeability are given by
er e 2:4a 0
mr m 2:4b 0
where m 4p 107 H=m is the permeability of vacuum and e 8:85 1012 F=m is the permittivity of vacuum.
With Eqs. (2.2) and (2.3), the wave equation can be derived for a source-free transmission line (or waveguide) or free space. For a source-free case, we have J r 0, and Eq. (2.2) can be rewritten as
H E 0 2:5a
H E jomH 2:5b
H H joeE 2:5c
H H 0 2:5d
...
- tailieumienphi.vn
nguon tai.lieu . vn